Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14)(15,16)(17,21)(18,19)(20,23)(22,24)(25,28)(26,27)(29,30)(31,37)(32,40)(33,36)(34,39)(35,41)(38,42)(43,44);(14,19)(16,23)(21,27)(24,28)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(5,6)(7,9)(8,10)(11,12)(13,15,18,20)(14,16,19,23)(17,25,26,22)(21,28,27,24)(29,30)(31,34)(32,40)(33,36)(35,41)(37,39)(38,43)(42,44);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,20)(15,19)(17,24)(18,23)(21,25)(22,27)(26,28)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(5,7,8,11)(6,9,10,12)(13,17)(14,21)(15,22)(16,24)(18,26)(19,27)(20,25)(23,28)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fh,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4296784);